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Hölder regularity for operator scaling stable random fields

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  • Biermé, Hermine
  • Lacaux, Céline

Abstract

We investigate the sample path regularity of operator scaling [alpha]-stable random fields. Such fields were introduced in [H. Biermé, M.M. Meerschaert, H.P. Scheffler, Operator scaling stable random fields, Stochastic Process. Appl. 117 (3) (2007) 312-332.] as anisotropic generalizations of self-similar fields and satisfy the scaling property where E is a dxd real matrix and H>0. In the case of harmonizable operator scaling random fields, the sample paths are locally Hölderian and their Hölder regularity is characterized by the eigen decomposition of with respect to E. In particular, the directional Hölder regularity may vary and is given by the eigenvalues of E. In the case of moving average operator scaling [alpha]-stable random fields, with [alpha][set membership, variant](0,2) and d>=2, the sample paths are almost surely discontinuous.

Suggested Citation

  • Biermé, Hermine & Lacaux, Céline, 2009. "Hölder regularity for operator scaling stable random fields," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2222-2248, July.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:7:p:2222-2248
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    References listed on IDEAS

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    1. Biermé, Hermine & Meerschaert, Mark M. & Scheffler, Hans-Peter, 2007. "Operator scaling stable random fields," Stochastic Processes and their Applications, Elsevier, vol. 117(3), pages 312-332, March.
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    Cited by:

    1. Kremer, D. & Scheffler, H.-P., 2019. "Operator-stable and operator-self-similar random fields," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 4082-4107.
    2. Li, Yuqiang & Xiao, Yimin, 2011. "Multivariate operator-self-similar random fields," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1178-1200, June.
    3. Didier, Gustavo & Meerschaert, Mark M. & Pipiras, Vladas, 2018. "Domain and range symmetries of operator fractional Brownian fields," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 39-78.
    4. Panigrahi, Snigdha & Roy, Parthanil & Xiao, Yimin, 2021. "Maximal moments and uniform modulus of continuity for stable random fields," Stochastic Processes and their Applications, Elsevier, vol. 136(C), pages 92-124.
    5. Sönmez, Ercan, 2018. "The Hausdorff dimension of multivariate operator-self-similar Gaussian random fields," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 426-444.
    6. Biermé, Hermine & Lacaux, Céline & Scheffler, Hans-Peter, 2011. "Multi-operator scaling random fields," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2642-2677, November.
    7. Dozzi, Marco & Shevchenko, Georgiy, 2011. "Real harmonizable multifractional stable process and its local properties," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1509-1523, July.
    8. Antoine Ayache & Geoffrey Boutard, 2017. "Stationary Increments Harmonizable Stable Fields: Upper Estimates on Path Behaviour," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1369-1423, December.
    9. Ayache, Antoine & Xiao, Yimin, 2016. "Harmonizable fractional stable fields: Local nondeterminism and joint continuity of the local times," Stochastic Processes and their Applications, Elsevier, vol. 126(1), pages 171-185.
    10. Ercan Sönmez, 2021. "Sample Path Properties of Generalized Random Sheets with Operator Scaling," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1279-1298, September.

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